Analytical Properties Of Ahlfors’ Constants In Covering Surface Theory

The Riemann sphere S is identified with the extended complex plane C via an ordinary stereographic projection.A covering surface ∑ is defined as a pair（f,（?））,where U is a Jordan domain in S and f is a non-constant holomorphic mapping from the closure U into S.For a covering surface ∑=（f,（?））,the area and perimeter of ∑ are defined as （?） respectively.The second fundamental theorem in Ahlfors covering surface theory is that,for any q（≥3）distinct points in C,there is a minimal positive constant H₀（a₁,…,a_q）（called the Ahlfors’ constant with respect to a₁,…,a_q）,such that the inequality （?） holds for each covering surface ∑=（f,（?））,where#denotes the cardinality.In particular,there is a minimal positive constant h0（a₁,…,a_q）（called the Ahlfors’ constant of exceptional-value cases with respect to a₁,…,a_q）,such that the inequality（q-2）A（∑）≤h₀（a₁,…,a_q）L（（?）∑）holds for each covering surface ∑=（f,U）with f（U）（?）\{a₁,…,a_q}.In other words,the second fundamental theorem in Ahlfors covering surface theory implies the existence of the Ahlfors’ constants H₀（a₁,…,a_q）and h₀（a₁,…,a_q）.It is difficult to compute H₀（a₁,…,a_q）and h₀（a₁,…,a_q）explicitly for general q distinct points a₁,…,a_q in C,and only few properties of them are known.We study the analytical properties of the Ahlfors’ constants in this thesis.We first prove the locally Lipschitz continuity and almost everywhere real-differentiability of H₀（a₁,…,a_q）and h₀（a₁,…,a_q）with respect to a₁,…,a_q,and consider the minima H₀（q）and h₀（q）of them for fixed q.Then,we discuss the further properties of H₀（a₁,a₂,a₃）for three distinct points a₁,a₂,a₃.We prove that H₀（a₁,a₂,a₃）is real-analytic outside a closed set with zero measure.